Indoor climbing

I used to work in a climbing shop selling shoes and I've been climbing over a decade, I'll give my 2p on the issue.
Tight shoes are necessary for performance but performance may not be necessary. A tight fit negates any movement in the toe, allowing more precision and the ability to stand on small footholds - a looser fit will have the shoe roll round the toe and slip off. If a shoe is fitted well, and downsized correctly, they will perform better for a longer period of their lifespan - if a shoe becomes too baggy as it stretches you will struggle to use it on smaller features or where more pressure/precision is needed. This may not actually be a problem at all depending on the type of climbing you are doing, or at a beginner level.

Another factor for comfort is the stiffness of the sole. A shoe with a harder and stiffer sole unit will support the foot more, requiring less effort to keep your toes on a small hold. And in the same way, a shoe with very soft rubber will support the foot less and may end up hurting your feet if you're spending a lot of time standing on small footholds as you have to work harder to keep your foot flat. Harder rubber tends to be more durable but less sticky and is less sensitive.
Shoes designed for all day comfort like this tend to be flat.

Shoes that have a downturn have a curved sole, are designed to force the toes into the end of the shoe for a tight fit. This helps transfer more force into the toes when standing on holds, but obviously decreases their comfort levels if you're standing in them all day. This isn't a problem if you're bouldering or sport climbing and are able to take the shoes off in between climbs and you aren't spending too long on your feet. Downturned shoes are suited to more overhanging climbing, with flat shoes being better for vertical and slab climbing.

They need to be tight in the right way, if they're tight because they don't suit the shape of your feet, they'll just be uncomfortable.
The fit should match the type of shoe. There's not much point in downsizing a pair of comfort shoes so they're mega tight (and uncomfortable), and no point in splashing £140 on some fancy aggressive shoes if they're not tight (IMO).
Getting used to climbing shoes is both important and necessary for improvement.
If you're just starting out, you'll probably benefit from having shoes that you can wear for longer because they're more comfortable rather than the negligible performance gain of a tighter shoe (for a beginner). You'll get better at climbing by being able to climb for longer.

Despite all this, it's still really hard to get it right and trial and error always comes into play - a frustrating prospect considering the cost of shoes. It's always a trade off of different factors and most climbers end up with different shoes for different purposes. Picking a pair of shoes to suit your needs is half the battle.

I'm breaking in a couple pairs of shoes at the moment - one of which I was on the verge of selling because I thought they were too tight but having endured with them I'm now really happy. It's taken ages and been painful. I've also only really started to get it right in the last few years (and I regret having not tried downsizing earlier) and I've found that shoes have a much better performance lifespan when fitted this way. I've also changed from exclusively wearing 5.10s to having a couple pairs of Scarpas. 5.10s seem to be made pretty cheaply these days, the last few pairs I've had have all fallen apart.

/csb